Peter Koellner: Large Cardinals Beyond Choice

The hierarchy of large cardinals provides us with a canonical means to
climb the hierarchy of consistency strength. There have been many
purported inconsistency proofs of various large cardinal axioms. For
example, there have been many proofs purporting to show that
measurable cardinals are inconsistent. But to date the only proofs
that have stood the test of time are those which are rather
transparent and simple, the most notable example being Kunen’s proof
showing that Reinhardt cardinals are inconsistent. The Kunen result,
however, makes use of AC, and long standing open problem is whether
Reinhardt cardinals are consistent in the context of ZF.

In this talk I will survey the simple inconsistency proofs and then
raise the question of whether perhaps the large cardinal hierarchy
outstrips AC, passing through Reinhardt cardinals and reaching far
beyond. There are two main motivations for this investigation. First,
it is of interest in its own right to determine whether the hierarchy
of consistency strength outstrips AC. Perhaps there is an entire
“choiceless” large cardinal hierarchy, one which reaches new
consistency strengths and has fruitful applications. Second, since the
task of proving an inconsistency result becomes easier as one
strengthens the hypothesis, in the search for a deep inconsistency it
is reasonable to start with outlandishly strong large cardinal
assumptions and then work ones way down. This will lead to the
formulation of large cardinal axioms (in the context of ZF) that start
at the level of a Reinhardt cardinal and pass upward through Berkeley
cardinals (due to Woodin) and far beyond. Bagaria, Woodin, and I
have been charting out this new hierarchy. I will discuss what we have
found so far.